hypograph

A hypograph of a function is a geometric object used in analysis and optimization. For a function f: X → R (or extended real-valued), the hypograph is the set hypo(f) = { (x, t) ∈ X × R : t ≤ f(x) }. The domain X is typically a subset of a topological space or vector space. The hypograph consists of all points lying on or below the graph of f.

As an example, if f(x) = x^2 on R, hypo(f) is the region in the plane consisting of all points (x, t) with t ≤ x^2; if f is constant c, hypo(f) is the half-space t ≤ c.

The epigraph of f is epi(f) = { (x, t) ∈ X × R : t ≥ f(x) }, the set of points on or above the graph. The hypograph is the reflection of epi(-f) across the horizontal axis. If f is upper semicontinuous, hypo(f) is closed; if f is lower semicontinuous, epi(f) is closed.

If f is concave, hypo(f) is a convex set; if f is convex, epi(f) is convex. These relationships are central to convex analysis and enable optimization techniques that rely on convexity.

Hypographs appear in the study of envelopes, in characterizing concave functions, and in variational analysis where functions can be represented or analyzed through their hypographs.